Knowing and Understanding©

Tim Whiteford PhD

We can know something, or we can know and understand it.

We know that the sun rises in the east. To understand what that means is quite another thing.

We can know that 3 x 6 = 18; to understand what that means is really what it's all about.

Knowing

We can look at knowing, or knowledge, as comprising two different sets or groups of knowledge; .conceptual knowledge and procedural knowledge. 

Knowledge is something we either have or do not have. We either know something or we do not know it.

Conceptual Knowledge is knowledge of ideas, concepts and relationships. Concepts are typically composed of many facets based on our experiences and how much we know about a particular idea. They can contain facts, ideas, feelings, relationships, senses, words, images, skills and so on. They usually occur in the form of schema and they are derived from the natural world. 

Here are some examples of conceptual knowledge typically found in math.

Angle, area, decimal, fraction, percentage, difference, division

What do you know about each one of these concepts? 

Procedural Knowledge is knowledge of the procedures, rules, formulae, symbols and special words we use in mathematics. Unlike conceptual knowledge, procedural knowledge is arbitrary knowledge  and a function of. 

Here are some example of procedural knowledge typically found in math.

Multiplication algorithm such as 12 x 8.

Symbols describing referents such as $, in., cm, lb

Fraction notation such as 1/2

Area of a particular shape such as l x b.

Understanding

Understanding is much more complex than knowing. It is much more difficult to state categorically (or assess) that you understand something or you do not understand it. We can also understand things at different levels depending upon our level of development and experience and familiarity with that particular thing.

We can look at understanding from two perspectives, both of which fulfill particular functions in our lives. They are instrumental understanding and relational understanding.   

Instrumental understanding is, for the most part  not good. It is isolated, fragile and disempowering. It is usually very tentative and difficult to act upon with any degree of confidence. It is at the opposite end of the continuum of understanding from relational understanding. It can be a rule without reason; a piece of rote memorization and can lead to the most amazing misconceptions. Sometimes, however, we need instrumental understanding just to "get by".

Relational Understanding is good. It is robust, connected, rich and full of interconnecting ideas. Relational understanding is constructed by the learner through  disequilibrium, assimilation and accommodation.

When we have relational understanding we are empowered and have confidence in our thinking. We can drive a car quite easily with just an instrumental understanding of how the car works. But what if it starts to make a loud knocking noise while driving home late at night. Do you stop immediately to avoid expensive repairs and risk being stranded on a dark lonely road late at night or do you drive home safe in the knowledge that the knocking sound is nothing more than a piece of wire wrapped around the axle.

Relational understanding can occur between pieces of conceptual knowledge, between pieces of conceptual knowledge and procedural knowledge and between pieces of procedural knowledge .

Wouldn't it be great if we had a relational understanding of everything. Is that possible? What is possible is that we should want to develop in our students the desire to have a relational understanding of those things that are important in our world.